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Electromagnetic quantities at a spacetime point have tensor Lorentz transformations between relatively-moving inertial frames. However, since the Lorentz transformation of time between inertial frames depends upon both the time and space coordinates, averages of electrodynamic quantities at a single time will in general depend upon the inertial frame, and will differ between inertial frames. Here we illustrate how the use of continuous charge and current distributions rather than point-charge distributions can lead to physically mystifying and even inaccurate results for electromagnetic quantities and physical phenomena. The discrepancy noted between the average electric field values in different inertial frames is particularly striking because it is first order in the relative velocity between the frames., Comment: 20 pages

A derivation of the Planck spectrum for thermal radiation is given based upon wave fluctuations within relativistic classical physics. The derivation depends crucially on thermal fluctuations existing above the fundamental inertial-frame-independent fluctuations of classical zero-point radiation. Such frame-independent zero-point fluctuations exist only in a relativistic wave theory and cannot exist in a nonrelativistic wave theory. Thus such a classical derivation of the Planck spectrum exists in a Lorentz-covariant classical theory, such as classical electrodynamics, but not in a Galilean-covariant theory where all waves are based upon material media. Classical zero-point radiation provides a purely classical alternative to quanta in the analysis of the Planck spectrum., Comment: 23 pages

A charged particle which is allowed to accelerate must have relativistic behavior because it is coupled to electromagnetic radiation which propagates at the speed of light. We treat the simple steady-state situation of a charged particle moving in a circular orbit with counter-propagating plane waves providing the power which balances the energy radiated away by the accelerating charge. It is emphasized that only an electromagnetic arrangement, such as a Coulomb potential or a constant magnetic field, can provide the relativistic central force for the particle motion., Comment: 14 pages

The interaction of a solenoid with a passing charged particle can be treated within classical or quantum physics. If charged particles pass around both sides of a solenoid, there is an experimentally-observed Aharonov-Bohm deflection of the double-slit particle interference pattern between charges passing on opposite sides. Such a deflection can be obtained by a classical force calculation. Although the magnitude of the angular deflection agrees between the classical force calculation and the quantum topological theory, the direction of the predicted deflection is opposite. Here we point out the simple basis for the direction of the deflection based upon classical electrodynamics and based upon quantum theory, and we mention analogues, both the electrostatic deflection of the particle interference pattern and the optical analogue of the classical calculation. The deflection direction involves an experimental question which is addressed rarely if ever., Comment: 12 pages

The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov-Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism., Comment: 16 pages

The Aharonov-Bohm phase shift in a particle interference pattern when electrons pass a long solenoid is identical in form with the optical interference pattern shift when a piece of retarding glass is introduced into one path of a two-beam optical interference pattern. The particle interference-pattern deflection is a relativistic effect of order 1/c^2, though this relativity aspect is rarely mentioned in the literature. Here we give a thorough analysis of the classical electromagnetic aspects of the interaction between a solenoid or toroid and a charged particle. We point out the magnetic Lorentz force which the solenoid or toroid experiences due to a passing charge. Although analysis in the rest frame of the solenoid or toroid will involve back Faraday fields on the charge, the analysis in the inertial frame in which the charge is initially at rest involves forces due to only electric fields where forces are equal in magnitude and opposite in direction. The classical analysis is made using the Darwin Lagrangian. We point out that the classical analysis suggests an angular deflection independent of Planck's constant where the deflection magnitude is identical with that given by the traditional quantum analysis, but where the deflection direction is unambiguous., Comment: 34 pages

Although the expressions for energy densities involving electric and magnetic fields are exactly analogous, the connections to forces and electromagnetic potentials are vastly different. For electrostatic situations, the changes in the \textit{electric} energy can be related directly to \textit{electric} forces and to the electrostatic potential. The situation involving magnetic forces and energy changes involves two fundamentally different situations. For charged particles moving with constant velocities, the changes in both electric and magnetic field energies are provided by the external forces that keep the particles' velocities constant; there are no Faraday acceleration electric fields in this situation. However, for particles that change speed, the changes in \textit{magnetic} energy density are related to acceleration-dependent Faraday \textit{electric} fields. Current undergraduate and graduate textbooks deal only with highly symmetric situations where the Faraday electric fields are easily calculated from the time-changing magnetic flux. However, in situations that lack high symmetry, such as the magnetic Aharonov-Bohm situation, the back (Faraday) acceleration electric fields of point charges seem unfamiliar. In this article, we present a simple unsymmetric example and analyze it using the Darwin Lagrangian. In \textit{all} cases involving changing velocities of the current carriers, it is the work done by the back (Faraday) acceleration \textit{electric} fields that balances the \textit{magnetic} energy changes., Comment: 15 pages

In the first quarter of the 20th century, physicists were not aware of the existence of classical electromagnetic zero-point radiation nor of the importance of special relativity. Inclusion of these aspects allows classical electron theory to be extended beyond its 19th century successes. Here we review spherical electromagnetic radiation modes in a conducting-walled spherical cavity and connect these modes to classical electromagnetic zero-point radiation and to electromagnetic scale invariance. Then we turn to the scattering of radiation in classical electron theory within a simple approximation. We emphasize that, in steady-state, the interaction between matter and radiation is disguised so that the mechanical motion appears to occur without the emission of radiation, even though the particle motion is actually driven by classical electromagnetic radiation. It is pointed out that, for nonrelativistic particles, only the harmonic oscillator potential taken in the low-velocity limit allows a consistent equilibrium with classical electromagnetic zero-point radiation. For relativistic particles, only the Coulomb potential is consistent with electrodynamics. The classical analysis places restrictions on the value of $e^{2}/\hbar c$., Comment: 29 pages

Energy equipartition is appropriate only for nonrelativistic classical mechanics, but has only limited relevance for a relativistic theory such as classical electrodynamics. In this article, we discuss harmonic-oscillator thermal equilibrium from three different perspectives. First, we contrast the thermal equilibrium of nonrelativistic mechanical oscillators (where point collisions are allowed and frequency is irrelevant) with the equilibrium of relativistic radiation modes (where frequency is crucial). The Rayleigh-Jeans law appears from applying a dipole-radiation approximation to impose the nonrelativistic mechanical equilibrium on the radiation spectrum. In this discussion, we note the possibility of zero-point energy for relativistic radiation, which possibility does not arise for nonrelativistic classical-mechanical systems. Second, we turn to a simple electromagnetic model of a harmonic oscillator and show that the oscillator is fully in radiation equilibrium (which involves all radiation multipoles, dipole, quadrupole, etc.) with classical electromagnetic zero-point radiation, but is not in equilibrium with the Rayleigh-Jeans spectrum. Finally, we discuss the contrast between the flexibility of nonrelativistic mechanics with its arbitrary potential functions allowing separate scalings for length, time, and energy, with the sharply-controlled behavior of relativistic classical electrodynamics with its single scaling connecting together the scales for length, time, and energy. It is emphasized that within classical physics, energy-sharing, velocity-dependent damping is associated with the low-frequency, nonrelativistic part of the Planck thermal radiation spectrum, whereas acceleration-dependent radiation damping is associated with the high-frequency adiabatically-invariant and Lorentz-invariant part of the spectrum corresponding to zero-point radiation., Comment: 31 pages

Here we review the understanding of the classical hydrogen atom in classical electromagnetic zero-point radiation, and emphasize the importance of special relativity. The crucial missing ingredient in earlier calculational attempts (both numerical and analytic) is the use of valid approximations to the full relativistic analysis. It is pointed out that the nonrelativistic time Fourier expansion coefficients given by Landau and Lifshitz are in error as the electromagnetic description of a charged particle in a Coulomb potential, and, because of this error, Marshall and Claverie's conclusion regarding the failure of radiation balance is invalid. Rather, using Marshall and Claverie's calculations, but restricted to lowest nonvanishing order in the orbital eccentricity (where the nonrelativistic orbit is a valid approximation to the fully relativistic electromagnetic orbit) radiation balance for classical electromagnetic zero-point radiation is shown to hold at the fundamental frequencies and associated first overtones., Comment: 31 pages

It is pointed out that an electric charge oscillating in a one-dimensional purely-harmonic potential is in detailed balance at its harmonics with a radiation bath whose energy $U_{rad}$ per normal mode is linear in frequency $\omega$, $U_{rad}=const\times\omega,$ and hence is Lorentz invariant, as seems appropriate for relativistic electromagnetism. The oscillating charge is NOT in equilibrium with the Rayleigh-Jeans spectrum which arises from energy-sharing equipartition ideas which are valid only in nonrelativistic mechanics. Here we explore the contrasting behavior of harmonic oscillators connected to baths in classical mechanics and electromagnetism. It is emphasized that modern physics text are in error in suggesting that the Rayleigh-Jeans spectrum corresponds to the equilibrium spectrum of random classical radiation, and in ignoring Lorentz-invariant classical zero-point radiation which is indeed a classical equilibrium spectrum., Comment: 21 pages

Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant $\hbar$) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution which arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant $\hbar$. All mechanical systems in thermal radiation will inherit the constant $\hbar$ from thermal radiation. Here we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions., Comment: 23 pages

Stochastic electrodynamics is the classical electrodynamic theory of interacting point charges which includes random classical radiation with a Lorentz-invariant spectrum whose scale is set by Planck's constant. Here we give a cursory overview of the basic ideas of stochastic electrodynamics, of the successes of the theory, and of its connections to quantum theory., Comment: 18 pages

It has been shown repeatedly over a period of 50 years that the use of relativistic classical physics and the inclusion of classical electromagnetic zero-point radiation leads to the Planck blackbody spectrum for classical radiation equilibrium. However, none of this work involves scattering calculations. In contrast to this work, currently accepted physical theory connects classical physics to only the Rayleigh-Jeans spectrum. Indeed, in the past, it has been shown that a nonlinear classical oscillator (which is necessarily a nonrelativistic scattering system) achieves equilibrium only for the Rayleigh-Jeans spectrum where the random radiation present at the frequency of the second harmonic of the oscillator motion has the same energy per normal mode as the radiation present at the fundamental frequency. Here we continue work emphasizing the importance of relativistic versus nonrelativistic analysis. We consider the scattering of random classical radiation by a charged harmonic oscillator of small but non-zero oscillatory amplitude (which can be considered as a relativistic scattering system) and show that detailed radiation balance holds not only at the fundamental frequency of the oscillator but through the first harmonic corresponding to quadrupole scattering, provided that the radiation energy per normal mode at the first harmonic is double the radiation energy per normal mode at the fundamental frequency. This condition corresponds exactly to the zero-point radiation spectrum which is linear in frequency. It is suggested that for this relativistic scattering system, the detailed balance for zero-point radiation holds not only for the fundamental and first harmonic but extends to all harmonics. Here we have the first example of the detailed balance of zero-point radiation under relativistic scattering.

In 1893, Wien applied the first two laws of thermodynamics to blackbody radiation and derived his displacement theorem. Believing that the information from thermodynamics had been exhausted, Planck turned to statistical ideas in 1900 in order to provide a physical understanding for his experimental--data-based interpolation giving the Planck spectrum without zero-point radiation. Here we point out that the third law of thermodynamics (which was developed in the early years of the 20th century) introduces additional thermodynamic information regarding thermal radiation. The Planck spectrum for thermal radiation can be derived from purely thermodynamic ideas applied to the classical simple harmonic oscillator, since every radiation mode takes a simple oscillator form. Thermodynamics alone implies the Planck spectrum including zero-point energy without any need for quantum theory or statistical ideas. It is pointed out that the inclusion of zero-point energy involves more natural thermodynamics than its exclusion., Comment: 22 pages

Students encounter harmonic-oscillator models in many aspects of basic physics, within widely-varying theoretical contexts. Here we highlight the interconnections and varying points of view. We start with the classical mechanics of masses coupled by springs and trace how the same essential systems are reanalyzed in the unretarded van der Waals interactions between dipole oscillators within classical and quantum theories. We note how classical mechanical ideas from kinetic theory lead to energy equipartition which determines the high-temperature van der Waals forces of atoms and molecules modeled as dipole oscillators. In this case, colliding heat-bath particles can be regarded as providing local hidden variables for the statistical mechanical behavior of the oscillators. Next we note how relativistic classical electrodynamical ideas conflict with the assumptions of nonrelativistic classical statistical mechanics. Classical electrodynamics which includes classical zero-point radiation leads to van der Waals forces between dipole oscillators, and these classical forces agree at all temperatures with the forces derived from quantum theory. However, the classical theory providing this agreement is not a local theory, but rather a non-local hidden-variables theory. The classical theory can be regarded as involving hidden variables in the random phases of the plane waves spreading throughout space which provide the source-free random radiation., Comment: 33 pages

Based upon thermodynamic ideas, two new derivations of the Planck blackbody spectrum are given within classical physics which includes classical zero-point radiation. The first and second laws of thermodynamics, applied to a harmonic oscillator or a radiation normal mode, require that the canonical potential $\phi(\omega/T)$ is a function of a single variable corresponding to the ratio of the oscillation frequency to the temperature. The second law of thermodynamics involves extremum ideas which may be applied to thermal radiation. Our first derivation of the Planck spectrum is based upon the idea that the canonical potential $\phi(\omega/T)$ is a monotonic function and all its derivatives are monotonic when interpolating between zero-point energy at low temperature and energy equipartition at high temperature; the monotonic behavior precludes the canonical potential from giving a preferred value for the ratio $\omega/T.$ Our second derivation of the Planck spectrum is based upon the requirement that the change in the Helmholtz free energy of the radiation in a partitioned box held at constant temperature should be a minimum at thermal equilibrium. Finally, the change in Casimir energy with change in partition position for the radiation in a partitioned box is shown to correspond at high temperature to the absence of zero-point energy when the spectral energy per normal mode is chosen as the traditional Planck spectrum which omits zero-point energy at low temperature; thus the idea of zero-point energy is embedded in the traditional Planck spectrum. It is emphasized that thermal radiation is intimately connected with zero-point radiation and the structure of spacetime in classical physics., Comment: 27 pages and 3 figures

We point out that current textbooks of modern physics are a century out-of-date in their treatment of blackbody radiation within classical physics. Relativistic classical electrodynamics including classical electromagnetic zero-point radiation gives the Planck spectrum with zero-point radiation as the blackbody radiation spectrum. In contrast, nonrelativistic mechanics cannot support the idea of zero-point energy; therefore if nonrelativistic classical statistical mechanics or nonrelativistic mechanical scatterers are invoked for radiation equilibrium, one arrives at only the low-frequency Rayleigh-Jeans part of the spectrum which involves no zero-point energy, and does not include the high-frequency part of the spectrum involving relativistically-invariant classical zero-point radiation. Here we first discuss the correct understanding of blackbody radiation within relativistic classical physics, and then we review the historical treatment. Finally, we point out how the presence of Lorentz-invariant classical zero-point radiation and the use of relativistic particle interactions transform the previous historical arguments so as now to give the Planck spectrum including classical zero-point radiation. Within relativistic classical electromagnetic theory, Planck's constant h appears as the scale of source-free zero-point radiation., Comment: 40 pages

We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum theories. In both theories, Planck's constant sets the scale of atomic phenomena. However, the roles played in the foundations of the theories are sharply different. In quantum theory, Planck's constant is crucial to the structure of the theory. On the other hand, in classical electrodynamics, Planck's constant is optional, since it appears only as the scale factor for the (homogeneous) source-free contribution to the general solution of Maxwell's equations. Since classical electrodynamics can be solved while taking the homogenous source-free contribution in the solution as zero or non-zero, there are naturally two different theories of classical electrodynamics, one in which Planck's constant is taken as zero and one where it is taken as non-zero. The textbooks of classical electromagnetism present only the version in which Planck's constant is taken to vanish., Comment: 9 pages

Here we discuss blackbody radiation within the context of classical theory. We note that nonrelativistic classical mechanics and relativistic classical electrodynamics have contrasting scaling symmetries which influence the scattering of radiation. Also, nonrelativistic mechanical systems can be accurately combined with relativistic electromagnetic radiation only provided the nonrelativistic mechanical systems are the low-velocity limits of fully relativistic systems. Application of the no-interaction theorem for relativistic systems limits the scattering mechanical systems for thermal radiation to relativistic classical electrodynamic systems, which involve the Coulomb potential. Whereas the naive use of nonrelativistic scatterers or nonrelativistic classical statistical mechanics leads to the Rayleigh-Jeans spectrum, the use of fully relativistic scatterers leads to the Planck spectrum for blackbody radiation within classical physics., Comment: 16 pages

A simple junior-level electrodynamics problem is used to illustrate the interference between a source-free standing plane wave and a wave generated by a pulse in a current sheet. Depending upon the relative phases between the standing wave and the current pulse and also upon the relative magnitudes, we can find quite different patterns of emitted energy and momentum. If the source gives a large radiation pulse so that the source-free plane wave can be neglected, then the radiation spreads out symmetrically on either side of the current sheet. However, if the radiation sheet gives a pulse with fields comparable to those of the standing wave, then we can find a single radiation pulse moving to the right while the current sheet recoils to the left, or the situation with the directions reversed. The example is a crude illustration of particle-like behavior arising from conventional classical electromagnetic behavior in the presence of source-free radiation. The discussion makes contact with the ideas of photons in modern physics., Comment: 15 pages

Here we show that within classical physics, the Planck blackbody spectrum can be derived directly from the structure of relativistic spacetime. In noninertial frames, thermal radiation at positive temperature is connected directly to zero-point radiation whose spectrum follows from the geodesic structure of the spacetime. The connection between zero-point radiation and thermal radiation at postive temperature is through a time-dilating conformal transformation in the noninertial frame. Transferring the spectrum back to Minskowski spacetime, the Planck spectrum is obtained., Comment: 10 pges

Electromagnetism is a \textit{relativistic} theory and one must exercise care in coupling this theory with \textit{nonrelativistic} classical mechanics and with \textit{nonrelativistic} classical statistical mechanics. Indeed historically, both the blackbody radiation spectrum and diamagnetism within classical theory have been misunderstood because of two crucial failures: 1)the neglect of classical electromagnetic zero-point radiation, and 2) the use of erroneous combinations of nonrelativistic mechanics with relativistic electrodynamics. Here we review the treatment of classical blackbody radiation, and show that use of Lorentz-invariant classical electromagnetic zero-point radiation can be used to explain both the Planck blackbody spectrum and Landau diamagnetism at thermal equilibrium within classical electromagnetic theory. The analysis requires that relativistic electromagnetism is joined appropriately with simple nonrelativistic mechanical systems which can be regarded as the zero-velocity limits of relativistic systems, and that nonrelativistic classical statistical mechanics is applied only in the low-frequency limit when zero-point energy makes no contribution., Comment: 24 pages

Today's textbooks of electromagnetism give the particular solution to Maxwell's equations involving the integral over the charge and current sources at retarded times. However, the texts fail to emphasize the role played by the choice of the boundary conditions corresponding to solutions of the homogeneous Maxwell equations. Here we discuss the role of these boundary conditions for an experimenter with a hypothetical charged harmonic oscillator as his equipment. We describe the observations of the experimenter when located near a radio station or immersed in thermal radiation at temperature T. The classical physicists at the end of the 19th century chose the homogeneous boundary conditions for Maxwell's equation based upon the experimental observations of Lummer and Pringsheim which measured only the thermal radiation which exceeded the random radiation surrounding their measuring equipment. Today at the beginning of the 21st century, classical physicists must choose the homogeneous boundary conditions for Maxell's equations to correspond to the full radiation spectrum revealed by the recent Casimir force measurements which detect all the radiation surrounding conducting parallel plates, including the radiation absorbed and emitted by the plates themselves. The random classical radiation spectrum revealed by the Casimir force measurements includes electromagnetic zero-point radiation, which is missing from the spectrum measured by Lummer and Pringsheim, and which cannot be eliminated by going to zero temperature. This zero-point radiation will lead to zero-point energy for all systems which have electromagnetic interactions. Thus the choice of the boundary conditions on the homogeneous Maxwell equations is intimately related to the ideas of zero-point energy and non-radiating ground states which are introduced in classes of modern physics., Comment: 19 pages

The physicists of the early 20th century were unaware of two ideas which are vital to understanding some aspects of modern physics within classical theory. The two ideas are: 1) the presence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity. In classes of modern physics today, the problem of blackbody radiation within classical physics is still described in the historical context of the early 20th century. However, the inclusion of classical zero-point radiation and of relativity now allows a completely satisfactory classical understanding of blackbody radiation with the Planck spectrum, as well as of some other aspects of modern physics. Here we sketch the current classical understanding of blackbody radiation, pointing out that thermodynamics allows the presence of classical zero-point radiation, and that use of nonrelativistic physics leads to the Rayleigh-Jeans spectrum while relativistic physics gives the Planck spectrum. The current textbooks of modern physics are a century out of date in presenting the connections between classical and quantum physics., Comment: 13 pages

The physicists of the early 20th century were unaware of two aspects which are vital to understanding some aspects of modern physics within classical theory. The two aspects are: 1) the presence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity. In classes in modern physics today, the problem of atomic collapse is still mentioned in the historical context of the early 20th century. However, the classical problem of atomic collapse is currently being treated in the presence of classical zero-point radiation where the problem has been transformed. The presence of classical zero-point radiation indeed keeps the electron from falling into the Coulomb potential center. However, the old collapse problem has been replaced by a new problem where the zero-point radiation may give too much energy to the electron so as to cause self-ionization. Special relativity may play a role in understanding this modern variation on the atomic collapse problem, just as relativity has proved crucial for a classical understanding of blackbody radiation., Comment: 15 pages

Whereas nonrelativistic mechanics always connects the total momentum of a system to the motion of the center of mass, relativistic systems, such as interacting electromagnetic charges, can have internal linear momentum in the absence of motion of the center of energy of the system. This internal linear momentum of the system is related to the controversial concept of "hidden momentum." We suggest that the term "hidden momentum" be abandoned. Here we use the relativistic conservation law for the center of energy to give an unambiguous definition of the "internal momentum of a system," and then we exhibit this internal momentum for the system of a magnet (modeled as a circular ring of moving charges) and a distant static point charge. The calculations provide clear illustrations of this system for three cases: a) the moving charges of the magnet are assumed to continue in their unperturbed motion, b) the moving charges of the magnet are free to accelerate but have no mutual interactions, and c) the moving charges of the magnet are free to accelerate and also interact with each other. It is noted that when the current-carrying charges of the magnet are allowed to interact, the magnet itself will contain internal electromagnetic linear momentum, something which has not been presented clearly in the research and teaching literature., Comment: 23 pages. This manuscript is related to arXiv:1408.3741, but has been thoroughly revised with a different focus

In this article, it is pointed out that Faraday induction can be treated from an untraditional, particle-based point of view. The electromagnetic fields of Faraday induction can be calculated explicitly from approximate point-charge fields derived from the Li\'enard-Wiechert expressions or from the Darwin Lagrangian. Thus the electric fields of electrostatics, the magnetic fields of magnetostatics, and the electric fields of Faraday induction can all be regarded as arising from charged particles. Some aspects of electromagnetic induction are explored for a hypothetical circuit consisting of point charges which move frictionlessly in a circular orbit. For a small number of particles in the circuit (or for non-interacting particles), the induced electromagnetic fields depend upon the mass and charge of the current carriers while energy is transferred to the kinetic energy of the particles. However, for an interacting multiparticle circuit, the mutual electromagnetic interactions between the particles dominate the behavior so that the induced electric field cancels the inducing force per unit charge, the mass and charge of the individual current carriers become irrelevant, and energy goes into magnetic energy., Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1408.3739

In this article, the self-inductance of a circular circuit is treated from an untraditional, particle-based point of view. The electromagnetic fields of Faraday induction are calculated explicitly from the point-charge fields derived from the Darwin Lagrangian for particles confined to move in a circular orbit. For a one-particle circuit (or for N non-interacting particles), the induced electromagnetic fields depend upon the mass and charge of the current carriers while energy is transferred to the kinetic energy of the particle (or particles). However, for an interacting multiparticle circuit, the mutual electromagnetic interactions between particles can dominate the behavior so that the mass and charge of the individual particles becomes irrelevant; the induced fields are then comparable to the inducing fields and energy goes into magnetic energy. In addition to providing a deeper understanding of self-inductance, the example suggests that the claims involving hidden mechanical momentum in connection with momentum balance for interacting multiparticle systems are unlikely to be accurate., Comment: 14 pages

It is pointed out that Coleman and Van Vleck make a major blunder in their discussion of the Shockly-James paradox by designating relativistic hidden mechanical momentum as the basis for resolution of the paradox. This blunder has had a wide influence in the current physics literature, including erroneous work on the Shockley-James paradox, on Mansuripur's paradox, on the motion of a magnetic moment, on the Aharonov-Bohm phase shift, and on the Aharonov-Casher phase shift. Although hidden mechanical momentum is indeed dominant for non-interacting particles moving in a closed orbit under the influence of an external electric field, the attention directed toward hidden mechanical momentum represents a fundamental misunderstanding of the classical electromagnetic interaction between a multiparticle magnet and an external point charge. In the interacting multiparticle situation, the external charge induces an electrostatic polarization of the magnet which leads to an internal electromagnetic momentum in the magnet where both the electric and magnetic fields for the momentum are contributed by the magnet particles. This internal electromagnetic momentum for the interacting multiparticle situation is equal in magnitude and opposite in direction compared to the familiar external electromagnetic momentum where the electric field is contributed by the external charged particle and the magnetic field is that due to the magnet. In the present article, the momentum balance of the Shockley-James situation for a system of a magnet and a point charge is calculated in detail for a magnet model consisting of two interacting point charges which are constrained to move in a circular orbit on a frictionless ring with a compensating negative charge at the center., Comment: 25 pages

A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces., Comment: 39 pages This paper has been withdrawn by the author due to a sign error in Eq. (75)

A model calculation using the Darwin Lagrangian is carried out for a magnet consisting of two current-carrying charges constrained by centripetal forces to move in a circular path in the presence of the electric field from a distant external point charge. In the limit that the magnet's two charges are non-interacting, the calculation recovers the only valid calculation for hidden mechanical momentum. However, if the magnet's charges are mutually interacting, then there is internal electromagnetic linear momentum associated with the perturbed magnet's electrostatic charge distribution and the motion of the magnet's charges. This internal electromagnetic momentum does not seem to be recognized as distinct from the familiar external electromagnetic momentum associated with the electric field of the external charge and the magnetic field of the unperturbed magnet. In the multiparticle limit, the hidden mechanical momentum becomes negligible while the internal electromagnetic momentum provides the compensating linear momentum required by the relativistic conservation law connecting the total linear momentum to motion of the center of energy. Whereas the changes in the external electromagnetic momentum are often associated with magnetic forces of order $1/c^{2},$ changes in the internal electromagnetic momentum are associated with the electrical forces of order $1/c^{2}$. These electrical forces are relevant to the Shockley-James paradox and to the experimentally observed Aharonov-Bohm and Aharonov-Casher phase shifts., Comment: 19 pages

Although Planck's constant h is currently regarded as the elementary quantum of action appearing in quantum theory, it can also be interpreted as the multiplicative scale factor setting the scale of classical zero-point radiation appearing in classical electromagnetic theory. Relativistic classical electron theory with classical electromagnetic zero-point radiation gives many results in agreement with quantum theory. The areas of agreement between this classical theory and Nature seem worth further investigation., Comment: 10 pages

An understanding of the interaction of a point charge and a magnetic moment is crucial for understanding the experiments involving electromagnetic momentum carried by permeable materials as well as the experimentally-observed Aharonov-Bohm and Aharonov-Casher phase shifts. Here we present two simple models for a magnetic moment which have vastly different interactions with a distant point charge. It is suggested that a satisfactory theoretical understanding of the interaction is still lacking and that the "hidden momentum" interpretation has been introduced into the textbook literature prematurely., Comment: 10 pages

Recently Mansuripur has called into question the validity of the Lorentz force in connection with relativistic electromagnetic theory. Here we present some very simple point-charge systems treated through order v^{2}/c^{2} in order to clarify some aspects of relativistic controversies both old and new. In connection with the examples, we confirm the validity of the relativistic conservation laws. The relativistic examples make clear that external forces may produce a vanishing torque in one inertial frame and yet produce a non-zero torque in another inertial frame, and that the conservation of angular momentum will hold in both frames. We also discuss a relativistic point-charge model for a magnetic moment and comment on the interaction of a point charge and a magnetic moment. Mansuripur's claims of incompatibiltiy between the Lorentz force and relativity are seen to be invalid., Comment: 23 pages

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature., Comment: 29 pages

Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. However, this zero-point radiation also leads to classical explanations for a number of phenomena which are usually regarded as requiring quantum physics. Here we provide a cursory overview of the classical electromagnetic theory which includes classical zero-point radiation, and we note the areas of agreement and disagreement between the classical and quantum theories, both of which contain Planck's constant h., Comment: 17 pages

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell's equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero-temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics., Comment: 21 pages

A relativistic classical field theory with zero-point radiation involves a vacuum corresponding to a scale-invariant spectrum of random classical radiation in spacetime with the overall constant chosen to give an energy (1/2)\hbar\omega per normal mode in inertial frames. Classical field theory with classical zero-point radiation gives the same field correlation functions as quantum field theory for the symmetrized products of the corresponding free massless fields in inertial frames; however, the interpretations in classical and quantum theories are quite different. Quantum field theory has photons in thermal radiation but not in the vacuum state; classical theory has radiation in both situations. The contrast in interpretations is most striking for the Rindler coordinate frame accelerating through zero-point radiation; classical theory continues tensor behavior over to the Rindler frame, whereas quantum theory introduces a new Rindler vacuum state. The classical interpretation of thermal behavior rests on two fundamental principles. i) A scale-invariant distribution of random radiation cannot correspond to thermal radiation at non-zero temperature. ii) A scale-invariant distribution of random radiation can acquire a correlation time which reflects the parameters of a spacetime trajectory through the scale-invariant radiation. Based on these principles, classical theory finds no basis for an accelerating observer to reinterpret zero-point radiation in terms of thermal radiation. In contrast, quantum field theory claims that an observer uniformly accelerated through zero-point flucturations of the Minkowski vacuum encounters a thermal bath at the temperature T=\hbar a/(2\pi ck)., Comment: 20 pages

Thermal scalar radiation in two spacetime dimensions is treated within relativistic classical physics. Part I involves an inertial frame where are given the analogues both of Boltzmann's derivation of the Stefan-Boltzmann law and also Wien's derivation of the displacement theorem using the scaling of relativitic radiation theory. Next the spectrum of classical scalar zero-point radiation in an inertial frame is derived both from scale invariance and from Lorentz invariance. Part II involves the behavior of thermal radiation in a coordinate frame undergoing (relativistic) constant acceleration, a Rindler frame. The radiation normal modes in a Rindler frame are obtained. The classical zero-point radiation of inertial frames is transformed over to the coordinates of a Rindler frame. Although for zero-point radiation the two-field correlation function at different spatial points at a single time is the same between inertial and Rindler frames, the correlation function at two different times at a single Rindler spatial coordinate is different, and has a natural extension to non-zero temperature. The thermal spectrum in the Rindler frame is then transferred back to an inertial frame, giving the familar Planck spectrum., Comment: 31 pages

The Planck spectrum of thermal scalar radiation is derived suggestively within classical physics by the use of an accelerating coordinate frame. The derivation has an analogue in Boltzmann's derivation of the Maxwell velocity distribution for thermal particle velocities by considering the thermal equilibrium of noninteracting particles in a uniform gravitational field. For the case of radiation, the gravitational field is provided by the acceleration of a Rindler frame through Minkowski spacetime. Classical zero-point radiation and relativistic physics enter in an essential way in the derivation which is based upon the behavior of free radiation fields and the assumption that the field correlation functions contain but a single correlation time in thermal equilibrium. The work has connections with the thermal effects of acceleration found in relativistic quantum field theory., Comment: 23 pages

It is pointed out that relativistic classical electron theory with classical electromagnetic zero-point radiation has a scaling symmetry which is suitable for understanding the equilibrium behavior of classical thermal radiation at a spectrum other than the Rayleigh-Jeans spectrum. In relativistic classical electron theory, the masses of the particles are the only scale-giving parameters associated with mechanics while the action-angle variables are scale invariant. The theory thus separates the interaction of the action variables of matter and radiation from the scale-giving parameters. Classical zero-point radiation is invariant under scattering by the charged particles of relativistic classical electron theory. The basic ideas of the matter -radiation interaction are illustrated in a simple relativistic classical electromagnetic example., Comment: 18 pages

The conservation laws of nonrelativistic and relativistic systems are reviewed and some simple illustrations are provided for the restrictive nature of the relativistic conservation law involving the center of energy compared to the nonrelativistic conservation law for the center of restmass. Extension of the nonrelativistic interaction of particles through a potential to a system which is Lorentz-invariant through order v^2/c^2 is found to require new velocity- and acceleration-dependent forces which are suggestive of field theory where the no-interaction theorem of Currie, Jordan, and Sudershan does not hold., Comment: 20 pages

It is shown that a nonrelativistic mechanical system involving a general nonrelativistic potential V(|r1-r2|) between point particles at positions r1 and r2 can be extended to a Lagrangian system which is invariant under Lorentz transformation through order v^2/c^2. However, this invariance requires the introduction of velocity-dependent and acceleration-dependent forces between particles. The textbook treatments of "relativistic mechanics" can be misleading; the discussions usually deal with only one particle experiencing prescribed forces and so make no mention of these additional velocity- and acceleration-dependent forces. A simple example for a situation analogous to a parallel-plate capacitor is analyzed for all the conservation laws of Galilean invariance or Lorentz invariance. For this system, Galilean invariance requires that the mechanical momentum is given by pmech=mv but places no restriction on the position-dependent potential function. On the other hand, Lorentz invariance requires that the mechanical momentum is given by pmech=mv/(1-v^2/c^2)^1/2, and in addition requires that the potential function is exactly the Coulomb potential V(|r1-r2|)=k/|r1-r2|. It is also noted that the transmission of the interparticle-force signal at the speed of light again suggests a special role for the Coulomb potential. A nonrelativistic particle system interacting through the Coulomb potential becomes the Darwin Lagrangian when extended to a system relativistic through order v^2/c^2, and then allows extension to classical electrodynamics as a fully Lorentz-invariant theory of interacting particles., Comment: 18 pages

One should not confuse a physical constant with a theory which incorporates the constant. Planck's constant h can appear in classical or quantum theories., Comment: 4 pages

The fact that the author of an excellent textbook on electromagnetism could be duped by "hidden momentum" vividly illustrates the problematic nature of its use., Comment: 4 pages

The long-standing controversy regarding the Aharonov-Bohm phase shift is reviewed. The shifts of both optical and particle interference patterns are summarized. It is pointed out that a line of electric dipoles and a line of magnetic dipoles (a long solenoid) both produce experimentally observed phase shifts similar to that produced by introducing a rectangular block of glass behind one slit of a double-slit interference pattern; the double-slit pattern is shifted while the single-slit envelope remains undisplaced. The quantum explanation for the magnetic interference pattern shift introduced by Aharonov and Bohm in 1959 involves completely different ideas from those suggested by a semiclassical analysis. Experiments planned by Caprez, Barwick, and Batelaan should clarify the connections between classical and quantum theories in connection with the Ahronov-Bohm phase shift., Comment: 10 pages

The interaction of a point charge and a magnetic moment (and by extension a point charge and a solenoid) is explored within well-defined point-charge magnetic-moment models where full calculations are possible. It is shown explicitly how the "hidden mechanical momentum" is introduced by the "hidden" external forces of constraint, requiring a prescribed response (through order 1/c^2) of the system to electromagnetic forces. These external forces often go unmentioned in the textbook and research literature. The dependence of "hidden mechanical momentum" upon detailed external (nonelectromagnetic) forces may undermine the idea's usefulness in describing nature. Some statements of dubious validity in the textbook literature are noted., Comment: 36 pages

Recent experiments undertaken by Caprez, Barwick, and Batelaan should clarify the connections between classical and quantum theories in connection with the Aharonov-Bohm phase shift. It is pointed out that resistive aspects for the solenoid current carriers play a role in the classical but not the quantum analysis for the phase shift. The observed absence of a classical lag effect for a macroscopic solenoid does not yet rule out the possibility of a lag explanation of the observed phase shift for a microscopic solenoid., Comment: 9 pages